This is somewhat related to this previous question I asked.
Let $K$ be a Kan complex, let $f,g:K\to K$ be morphisms of simplicial sets. Consider a simplex $\sigma\in K_n$, and let $\alpha$ be a vertex of $\sigma$. Assume I have a $1$-simplex $\gamma\in K_1$ such that $\partial_0\gamma = f(\alpha)$ and $\partial_1\gamma = g(\alpha)$. I will describe a way to construct a morphism $$h_\sigma\in\hom(\Delta^1\times\Delta^n,K)$$ such that the restrictions of $h_\sigma$ to $\{0\}\times\Delta^n$ and $\{1\}\times\Delta^n$ correspond to the simplices $f(\sigma)$ and $g(\sigma)$ respectively from this data. I will implicitly use the well known fact that $$\hom(\Delta^p,K)\cong K_p$$ (coming from the Yoneda lemma) in what follows.
First, we give an explicit description of $\Delta^1\times\Delta^n$. Consider the poset $$P_n = \{[i,j]\mid i\in\{0,1\},0\le k\le n\}$$ with $[i_1,j_1]<[i_2,j_2]$ if, and only if $i_1\le i_2$, $j_1\le j_2$, and at least one of the two inequalities is strict. For example, if $n=2$ the poset can be represented by $$\require{AMScd} \begin{CD} [0,1] @>>> [1,1] @>>> [2,1]\\ @AAA @AAA @AAA\\ [0,0] @>>> [1,0] @>>> [2,0] \end{CD}$$ Then non-degenerate $k$-simplices in $\Delta^1\times\Delta^n$ are represented by strictly increasing $k$-tuples in $P_n$, and faces are obtained by deleting an element in a tuple. For details, see for example here (page 45).
Without loss of generality, let $\alpha$ be the $0$-th vertex of $\sigma$. We construct $h_\sigma$ by describing it on non-degenerate simplices, using the data we have as starting point and relying heavily on the horn-filling property. Therefore, a maximal non-degenerate simplex will be an $(n+1)$-tuple of the form $$([0,0],[1,0],\ldots,[i,0],[i,1],\ldots,[n,1]).$$ We order these simplices depending on the $i$.
Of course, we define \begin{align} h_\sigma(([0,0],\ldots,[n,0])) = & f(\sigma),\\ h_\sigma(([0,1],\ldots,[n,1])) = & g(\sigma),\\ h_\sigma(([0,0],[0,1])) = & \gamma. \end{align} Now consider the first of the maximal simplices, given by $$([0,0],[0,1],\ldots,[n,1]).$$ Notice that we have $h_\sigma(([0,0],[0,1]))$ and $h_\sigma(([0,1],[k,1]))$ for all $k$ (given by a $1$-simplex in $g(\sigma)$). But this is exactly the data of a horn of the simplex $([0,0],[0,1],[k,1])$, so we get a filler for all these simplices thanks to the fact that $K$ is a Kan complex. Using this data, we can similarly fill $3$-simplices, then $4$-simplices, and so on, up to filling the whole $(n+1)$-simplex.
Some of the data thus obtained allows us to start a similar process for the second $(n+1)$-simplex, filling it, and so on up to filling the whole $\Delta^1\times\Delta^n$. I can make this more precise if needed.
Questions:
1) Has something similar already been done? Well, I'm pretty sure that the answer to this one is yes, but does anybody have a nice reference?
2) Can this be used to construct a (simplicial) homotopy between $f$ and $g$ if we are given a $\gamma$ for each $0$-simplex in $K$? (I think we might have some kind of "functoriality" problems here. Namely, when we do our filling process we obtain the $1$-simplices $([i,0],[i,1])$, which might not agree with the corresponding $\gamma$. Is there a way to solve this problem?)
Edit: Crossposted to MO, where it was answered.